Abstract

Discrete transformations are widely used in the fields of signal and image
processing. Applications in the areas of data compression, template matching, signal
filtering pattern recognition all utilise various discrete transforms. The calculation of
transformations is a computationally intensive task which in most practical
applications requires considerable computing resources. This characteristic has
restricted the use of many transformations to applications with smaller datasets or
where real-time performance is not essential.
This restriction can be removed by the application of parallel processing
techniques to the calculation of discrete transformations. The aim of this thesis is to
determine efficient parallel algorithms and processor topologies for the
implementation of the discrete Walsh, cosine, Haar and D4 Daubauchies transforms,
and to compare the operation of the parallel algorithms running on T800 Transputers
with the equivalent serial von Neumann type algorithm. This thesis also examines the
transformations of a number of test functions in order to determine their ability to
represent various common global and locally defined functions.
It was found that the parallel algorithms developed during the course of this
thesis for the discrete Walsh, cosine, Haar and D4 Daubauchies transforms could all
be efficiently implemented on a hypercube processor topology.
Development of a number of parallel algorithms also led to the discovery of a
new parallel algorithm for the calculation of any transformation which can be
expressed as a Kronecker or tensor product/sum. A hypercube based algorithm was
devised which converts the Kronecker product to a Hadamard product on a hypercube
structure. This provides a simple algorithm for parallel implementations.
Examination of the four sets of transform coefficients for the test functions
revealed that all the transforms examined were not suitable for representing functions
with large numbers of discontinuity's such as the chirp function. Also, transforms with
local basis functions such as the Haar and D4 Daubauchies transforms provided better
representations of localised functions than transforms consisting of global basis
function sets such as the discrete Walsh and cosine transformations.